Narrow Results

Let $R$ be a commutative Noetherian ring of dimension $d$. First, we define the “geometric subring” $A$ of a polynomial ring $R[T]$ or $R[T,T^{-1}]$ of dimension $d+1$ (the definition of geometric subring is more general, see (1.2)). Then we prove that every locally complete intersection ideal of height $d+1$ is a complete intersection ideal. Thus, improving the general bound of Mohan Kumar [On two conjectures about polynomial rings, Invent. Math.46(3) (1978) 225–236] for an arbitrary ring of dimension $d+1$. Afterward, we deduce that every finitely generated projective $A$-module of rank $d+1$ splits off a free summand of rank one. This improves the general bound of Serre [J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle. Séminaire Dubreil. Algébre et théorie des nombres, Volume 11 (1957–1958) no. 2, Talk No. 23, p. 18] for an arbitrary ring. Finally, applications are given to a set-theoretic generation of an ideal in the geometric ring $A$ and its polynomial extension $A[X]$.

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces ${\mathbb{P}}^n({𝒯})$ constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra $C({\mathbb{P}}^1({𝒯}))$ realized as a concrete groupoid C*-algebra, and find its $K$-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra $C({\mathbb{P}}^1({𝒯}))$, we identify those quantum principal $U(1)$-bundles introduced by Hajac and collaborators among the projections classified.

We deconstruct the finite projective modules for the fuzzy four-sphere, described in a previous paper, and correlate them with the matrix model approach, making manifest the physical implications of noncommutative topology. We briefly discuss also the U(2) case, being a smooth deformation of the celebrated BPST SU(2) classical instantons on a sphere.

We show how to generalize our method, based on projective modules and matrix models, which enabled us to derive noncommutative monopoles on a fuzzy sphere, to the non-Abelian case, recovering known results in literature. We then discuss a possible candidate for deforming the commutative Chern class to the noncommutative case.

Let $G$ be a finite simple group of Lie type. In this paper, we study characters of $G$ that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of $G$. Such characters can be viewed as a natural generalization of the Steinberg character. For groups $G$ of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

We study some applications of the theory of commutative monoids to the monoid of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

Let L = X(m; n), X ∈ {W, S, H, K}, be a graded simple Lie algebra of Cartan type over an algebraically closed field of characteristic p > 3. Then L is a so-called generalized restricted Lie algebra. Let be the primitive p-envelope of L, and G = X(m; 1), a subalgebra of . In this paper, a close connection between Cartan invariants for and U(G, χ) is established, where χ ∈ L* is extended to be a linear function on trivially, and 1 ≤ ht(χ) < p-2+δ_XW. This reduces the study of projective representations of the generalized restricted Lie algebra L to the one of the corresponding restricted Lie algebra G. As a special case, we recover some results in [Shu and Jiang, On Cartan invariants and blocks of Zassenhaus algebras, Comm. Algebra33(10) (2005) 3619–3630].

Given modules M and N, M is said to be N-subprojective if for every epimorphism g : B → N and homomorphism f : M → N, there exists a homomorphism h : M → B such that gh = f. For a module M, the subprojectivity domain of M is defined to be the collection of all modules N such that M is N-subprojective. As an alternative perspective on the projectivity of a module, a module M is said to be p-indigent if its subprojectivity domain is smallest possible, namely, consisting of exactly the projective modules. Properties of subprojectivity domains and of p-indigent modules are studied. For various classes of modules (such as simple and singular), necessary and sufficient conditions for the existence of p-indigent modules of those types are studied. We characterize the rings over which every (simple) module is projective or p-indigent. In addition, we use our results to provide a characterization of a special class of QF-rings in which the subinjectivity and subprojectivity domains of all modules coincide. As the projective analog of indigent modules, p-indigent modules were introduced by Holston, López-Permouth, Mastromatteo and Simental-Rodriguez. The paper is inspired by similar ideas and problems in papers by Aydoğdu and López-Permouth and by Alizade, Büyükaşık and Er, where an injective version of p-indigent modules is introduced and studied.

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory.

Let $R$ be a commutative Noetherian local ring. A finitely generated Gorenstein projective $R$-module $M$ is called IG-projective if there exists either an irreducible epimorphism $P\to M$ or irreducible monomorphism $M\to P$ with $P$ being a projective $R$-module. In this paper, we give a note on Gorenstein Projective Conjecture: a finitely generated IG-projective module is projective if and only if it is self-orthogonal.

Let $𝔤$ be the special linear Lie algebra ${𝔰𝔩}_3$ of rank 2 over an algebraically closed field $k$ of characteristic 3. In this paper, we classify all irreducible representations of $𝔤$, which completes the classification of the irreducible representations of ${𝔰𝔩}_3$ over an algebraically closed field of arbitrary characteristic. Moreover, the multiplicities of baby Verma modules in projective modules and simple modules in baby Verma modules are given. Thus we get the character formulae for simple modules and the Cartan invariants of $𝔤$.

A module $M$ is called coseparable ($𝔪$-coseparable) if for every submodule $U$ of $M$ such that $M/U$ is finitely generated ($M/U$ is simple), there exists a direct summand $V$ of $M$ such that $V\subseteq U$ and $M/V$ is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ($𝔪$-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of $𝔪$-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ($𝔪$-coseparable).

In the hyperalgebra ${{𝒰}}_r$ of the $r$th Frobenius kernel ${({\mathrm{\text{SL}}}_2)}_r$ of the algebraic group ${\mathrm{\text{SL}}}_2$, we construct a basis of the ${{𝒰}}_r$-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the ${{𝒰}}_r$-modules and the algebra ${{𝒰}}_r$.

Let $R$ be a ring of dimension $d\ge 1$ containing $\mathbb{Q}$ and $A=\frac{R[X,Y,Z,W]}{(XY-ZW)}$. This paper examines the question of efficient generation of ideals in $A$.

In the last decade, two new approaches have been introduced for the analysis of the projectivity of modules. In this paper, the projectivity of modules has been studied with a new approach by using projectively generated proper classes. As an opposite to projectivity, a module $M$ is said to be $\pi$-indigent if the projectively generated proper class by $M$ is consisting of exactly the split short exact sequences. We consider rings over which every (finitely generated, finitely presented, cyclic, simple) right $R$-modules are either projective or $\pi$-indigent.

Let $𝔤$ be a basic classical restricted Lie superalgebras over an algebraically closed field $k$ of odd characteristic $p$, and $U_0(𝔤)$ be the restricted enveloping algebra of $𝔤$. The purpose of this paper is to study the representations of $U_0(𝔤)$. First, we prove a $U_0(𝔤)$-module $Q$ is projective if and only if it has both a baby Verma module filtration and a twist baby Verma module filtration. Second, we get the Humphreys–Brauer type reciprocity on projective module, baby Verma module and irreducible module.